@Zaljohar: "Size Theory:
Size Theory is an extension of  ZFC,
having two additional primitives:
  S ( denoting Size) which is a 1-place function symbole.
and < ( symbolizing 'lesser than' ) , which is a binary relation.
Definition:  Sa > Sb <-> Sb < Sa
from axioms of identity we have the following theorems:
Sa=Sa
Sa=Sb <-> Sb=Sa
Sa=Sb=Sc -> Sa=Sc
AxE!y( y=S(x) )
Axioms:-
Axiom:    Aab ( Sa < Sb xor Sa=Sb xor Sa > Sb ).
Axiom: ~ Sa < Sa
Axiom:    Sa < Sb < Sc  -> Sa < Sc
from the above we have the following theorems:
Sa < Sb -> ~Sb < Sa
Sa < Sb <-> (~ Sa > Sb & ~ Sa=Sb )
Sa = Sb <-> (~ Sa > Sb & ~ Sa<Sb )
Sa > Sb <-> ( ~Sa < Sb & ~ Sa=Sb )
~Sa < Sb <-> (Sa > Sb xor Sa=Sb)
~Sa = Sb <-> (Sa > Sb xor Sa<Sb)
~Sa > Sb <-> (Sa < Sb xor Sa=Sb)
Axiom: SSa=Sa
Definition: a is Z-homomorphic_to b iff
 ( (every injection a->b is a bijection) or
   (no bijection exists between a and b) ) .
If a is Z-homomorphic_to b, we have the following axioms:
Axiom: Sa = Sb <-> Ef(f:a->b, f is bijective)
Axiom: Sa < Sb <-> Ef(f:a->b, f is injective,~ f is surjective)
Axiom: Sa > Sb <-> Ef(f:a->b, f is injective,~ f is surjective)
The followings are two theorems of this theory.
card x < card y <-> ( Sx < Sy &   x is Z-homomorphic_to y ).
card x = card y <-> ( Sx = Sb v ~ x is Z-homomorphic_to y ).
Axiom: AaAb( (a subset_of b & ~a=b) -> Sa < Sb )
Axiom: For any c,a,b,z,x,y  were a disjoint b and x disjoint y
 and c=aUb and z=xUy and Sy=Sb then:
 Sa=Sx<->Sc=Sz
 Sa<Sx<->Sc<Sz
Definition of gap: For any infinite subset b of w orderd in an
ascending manner after the set sizes of its members ,i.e. the member
with lesser size Preceeds the member with larger size.
then a gap in b is defined as the set of all members of w that
intervene between any two immediately successive members of b.
gaps will be denoted as g0,g1,g2,g3,.... of b.
g0 of b = { x| xew & ~x=0 & Sx<Sm1 & m1 is the first member in b }.
g1 of b = { x| xew & Sm1<Sx<Sm2 & m2 immediately succeed m1 in b }
In general:
gi of b = { x| xew & Smi<Sx<Sm(i+1) & m(i+1) immediately succeed mi in
b }.
example: b= { 0,2,4,6,....}
have g0={ }, g1={1} , g2={3} ,.....
Definition of gap sequence:
gap sequence 'gs' of b is the sequence of all gaps in b
gs(b) = g0,g1,g2,g3,........
for the example above:
gs(b) = {},{1},{3},{5},.......
the gap size sequence 'gss' is the sequence of sizes of gaps in b
gss(b)= Sg0,Sg1,Sg2,Sg3,.......
for the above example:
gss(b)=0,1,1,1,1,1,1,.........
Axiom: For any two subsets of w: a , b :
gss(a)=gss(b) -> Sa = Sb
Example: b={0,2,4,6,...} and a= { 1,3,5,...}
here we have b=a since both have the same gss that is
gss(a)=gss(b)= 0,1,1,1,1,1,.......
Axiom: for any two subsets of w: a,b :
if for every i , Sgi of a > Sgi of b then
Sa < Sb
Axiom: for any two subsets of w: a,b:
if a and b have repeated gap size sequence , then if the average gap
size sequence of a is bigger than that of b then
Sa < Sb
Example: if gss(a)=  1,2,3,1,2,3,1,2,3,.............
and gss(b)= 3,3,3,3,......
then Sb < Sa
since the average of gap sizes of a is (1+2+3)/3 =2
while that of b is 3.
( Note: this method of defining gaps can be extended by adding axioms
that define it for higher ordinals than w )
This gap size sequence method can define Division, Subtraction
and Root extraction of Sizes of some infinite sets, While Cantorian
cardinality cannot do that.
Axiom: For every a and b
Ef(f:a->b, f(x) = -x , f is bijective) -> Sa=Sb
Axiom: For every a and b
Ef(f:a->b, f(x)=1/x , f is bijective) -> Sa=Sb
/
Theory definition finished.
However one can always consistently add axioms to this theory to
define another theory that determine the size of sets of interest.
Zuhair"
    @Zaljohar: "<quote>
What you said is not altogether correct, for example this theory
can define size comparisons between any two sets that are
Z-homomorphic, and this is not only the natural numbers as you think.
Yes
Yes
The problem is with the case of non z-homomorphism, it is in this
situation were indeterminism of set size comparison lies.
I am thinking of a way of selecting an injective function from between
all the injective functions existing between a two non homomorphic
sets that can determine the size comparison between them.
For instance what if we define the injective function from x to y
containing the minimal ordered pair for each member of x as the set
size determining function.
What I mean by a minimal ordered pair in  injective function from x to
y
 for each member of x , is the pair <r s> rex and sey such that | |r|-|
s| | is the minimal
and if two pairs have the same | |r|-|s| |for example <r1 s1>
and <r2 s2> such that ||r1|-|s1||=||r2|-|s2|| then if |s1| < |s2|
then the pair <r1 r2> is considered a minimal ordered pair in the
injective function from x to y that has r1 being a member of x.
So the injective function from x to y that contain the minimal ordered
pair for each r in x, would be the one selected as the minimal
injective function and thus the set size determining function.
And then we determine the size comparison of x and y as follows:
1) If the minimal injective function from x to y is bijective then
Sx=Sy
2) if the minimal injective function from x to y is injective and not
surjective then
Sx<Sy
Now of course I am not sure of that at all, I only intend to apply to
the ordinals. But if it proves consistent on the ordinals then perhaps
it might be extended to involve integers, rationals and reals as well.
Zuhair"
    @Zaljohar: "<quote>
This is wrong: the set of odd numbers doesn't have all gaps=1
the gap size sequence of the set of odd numbers O is:
gss(O)= 0,1,1,1,1,......
 is "bigger" than
What is this set? what is the first number in this set?
what is the value of g0 for this set?
however since you say that all gaps=2, then definitly the set of all
odds is bigger in cardinality than this set.
I don't know why the question mark ???"
    @google: "<quote>
The set contains all numbers except that (very sparingly, but
infinitely often) two consecutive numbers are dropped.
You started your size theory with the idea that it be closer
to intuitive notions of set size than cardinality.
But saying that {1,3,5,7,...} is bigger than any set which has
all gaps bigger than 1, no matter how rare the gaps are,
seems at least counter-intuitive.
Especially, not even density of subsets of N is a
valid indicator of set size (density of odds: 1/2,
density of my other set: 1-2/10^100)."
    @Zaljohar: "<quote>
NO of course, not. I actually didn't know what is the set
you are talking about. See the average gap size axiom.
You are underestimating this system.
I don't really know what you are talking about here
Is there a repeated gap size pattern for this set you are
talking about, Is the avergae of this gap pattern
known, if so it looks to me that the averge
gap size is bigger in the case of odds than
in the case of the later set of yours,
so if this is the case then the set of odds would be
smaller in size than the mod set you
mentioned.
Zuhair
- Hide quoted text -"
    @jazzmobe: "<quote>
No, it is not "no doubt" since the notion of a theory being "superior"
to another theory is NOT merely that the one theory is richer than the
other.
MoeBlee"
    @Zaljohar: "<quote>
This theory is BOTH superior (i.e. has ZFC as a subtheory of it) and
richer than the traditional ways of thinking. I have no doubt in that.
But the main problem with this theory is that it is not complete, or
lets say it only define size for a limited class of sets. However I
presented this theory here not as a final theory on size, but rather
as a preliminary step towards extending it so that it can define size
for all sets of interest in mathematics and logics.
Zuhair"
    @jazzmobe: "<quote>
YOU say "i.e." that a theory is superior on account of the other
theory being a subtheory, as you simply missed my point that we are
NOT required to accept that "i.e."
MoeBlee"
    @Zaljohar: "<quote>
I didn't say a theory is superior on account of the other theory being
a subtheory. There is no 'on account' here. you have a meaning of
superior to be something like better, while what I mean by superior IS
that the other theory is a subset of it. I have got lost in
translation here.
Let me omit the word superior since it caused all this confusion.
Let me rewrite it again: this theory has ZFC as subtheory of it
and it is richer than the cardinality approach of Cantor's.
and what I mean by richer is that it both have all the virtues of
Cantor's approach and on top of it it can define finer set sizes.
Zuhair"
    @jazzmobe: "<quote>
Okay, I'll accept that, if your theory has ZFC embedded in it and also
expresses other relations.
MoeBlee"
    @jesse: "<email_address> (Daryl McCullough) writes:
<quote>
You seem to have missed his second disjunct.
ZFC proves
 a is Z-homomorphic_to b 
 <-> (a is finite and b is finite) -or-
     NOT (|a| = |b|)
"After years of arguing I realize that your intellects are too limited
to fully grasp my work.  [...] Still, no matter how child-like your  
minds are, [...] since you have language, [...] there's a chance that 
I'll be able to find something that your minds can handle." JSH"
    @Zaljohar: "On Jul 18, 11:13 am, <email_address> (Daryl McCullough)
wrote:
<quote>
No, that is not correct.
In ZFC if a is Z-homomorphic_to b then
this entails the following
(a is finite OR b is finite) xor (a is infinite and b is infinite and
no bijection exist between a and b).
so for example, w is Z-homomorphic_to Pw.
Zuhair"
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